Growth Models: A Practical Guide

1
Growth ModelsA Practical Guide

• Sarah O. Meadows
• Center for Research on Child Wellbeing
• Princeton University
• October 15, 2007

2
Outline of Presentation

• What are growth models?
• Nuts and Bolts
• Hands-On Example
• Additional Issues

3
Part I What is a Growth Model?
4
What is a Growth Model?

• A way to assess individual stability and change,
  both growth and decay, over time.
• A two-level, hierarchical model that that models
  (1) within individual change over time and (2)
  between individual differences in patterns of
  growth.

5
A Rose by Any Other Name . . .

• Growth Models
• Trajectory Models
• Growth Curve Models
• Latent GM
• Latent TM
• Latent GCM
• Hierarchical Models

• Random Intercept Models
• Random Coefficient Models
• Random Intercept/Random Slope Models
• Variance Component Models

6
Why Latent?

• Because we assume that whatever process that is
  underlying the thing we are modeling (or the
  behavior we observe) is actually unobserved, or
  latent.
• The characteristics we observe are a
  manifestation of this latent trajectory.
• This language grew out of structural equation
  modeling (SEM).

7
Why use GMs?

• Everyone else is doing it!
• Education
• Criminology
• Psychology
• Sociology
• Public Health
• You have longitudinal data and are interested in
  change over time.
• You may want to explain those changes.
• You may also believe that not everyone follows
  the same path.

8
How Have Others Used GMs?

• Growth Trajectories of Sexual Risk Behavior in
  Adolescence and Young Adulthood. Fergus,
  Zimmerman, Caldwell. American Journal of
  Public Health. 2007.
• Individual Differences in the Onset of Tense
  Marking A Growth Model Example. Hadley
  Colt. Journal of Speech, Language, and Hearing
  Research. 2006.
• Ten-Year Stability of Depressive Personality
  Disorder in Depressed Outpatients. Laptook,
  Klein, Dougherty. The American Journal of
  Psychiatry. 2006.
• Verbal Learning and Everyday Functioning in
  Dementia An Application of Latent Variable
  Growth Curve Modeling. Mast Allaire. The
  Journals of Gerontology. 2006.
• You Make Me Sick Marital Quality and Health
  Over the Life Course. Umberson, Williams,
  Powers. Journal of Health and Social Behavior.
  2006.
• Parental Divorce and Child Mental Health
  Trajectories. Strohschein. 2005. Journal of
  Marriage and Family.

9
A Detailed Example

• Stability and Change in Family Structure and
  Maternal Health Trajectories. Meadows,
  McLanahan, Brooks-Gunn. American Sociological
  Review. Forthcoming.
• We wanted to know whether changes in family
  structure, including transitions into and out of
  coresidential relationships, had short-term
  impacts on health (i.e., crisis model) or
  long-term impacts on health (i.e., resource
  model).

10
Example (cont.)

• Trajectories of maternal self-rated health and
  mental health problems from one year after birth
  to five years after birth.
• Two measures of family structure change
• Level 1 Time-Varying
• Level 2 Time-Invariant

11
Example (cont.)

• Results
• Transitions, especially exits from marriages,
  resulted in short-term declines in physical
  health and short-term increases in mental health
  problems.
• Little support for the resource model no growing
  gap in well-being between mothers who remained
  stably married and those remained stably single,
  as well as mothers who made transitions.

12
Figure 1. Mothers Mental Health Trajectories
13
Figure 2. Mothers Household Income Trajectories
14
Figure 3. Fathers Mental Health Trajectories
15
Figure 4. Fathers Earnings Trajectories
16
Part II Nuts and Bolts
17
Where Did GMs Come From?

• Time Series Models (Autoregressive)
• Repeated Measures ANOVA
• (Duncan Duncan, 2004)
• SEM
• Multilevel Models (HLM)

18
Hierarchical Models

• Traditional
• Level 1 Students
• Level 2 Schools
• Growth Models (a type of HM)
• Level 1 Repeated Observations
• Level 2 Individuals

19
Unconditional Model

• Level 1 Within Individual
• Level 2 Between Individual

20
A Latent Trajectory
Latent Depression Trajectory
ß
Depressive Symptoms
a
Time
21
Time-Invariant Covariates

• Level 1 Within Individual
• Level 2 Between Individual

22
Time-Varying Variables

• Level 1 Within Individual
• Level 2 Between Individual

Time-varying effect.
23
Fixed vs. Random

• Fixed Means of the latent trajectory parameters
  (i.e., intercept and slope)
• Random Variance of the latent trajectory
  parameters (i.e., indicates individual
  heterogeneity around population means)

24
Part III An Example
25
Software

• MPlus SEM based
• HLM Hierarchical Modeling
• SAS Proc Traj
• STATA

26
Data Requirements

• Three observations
• For polynomial curves you need d 2 repeated
  measures, where d is the degree of the
  polynomial.
• Horizontal data file (i.e., one person, one row).
• Convert data to .dat file.
• Remember the order of the variables!!

27
Self-Rated Health

• Mothers in FFCWS
• In general, how is your health?
• Excellent (5)
• Very Good (4)
• Good (3)
• Fair (2)
• Poor (1)
• Repeated measures one, three, and five years
  after birth.

28
Setting the Trajectory
Intercept
Slope
1
0
4
1
1
2
SRH 1
SRH 3
SRH 5
29
Models

• Unconditional
• Model Fit
• Conditional
• Time-Invariant Covariates
• MPlus Graphs
• Selection and Causation
• Time-Varying Covariates

30
Model Fit

• Chi-Square
• Not Significant, but almost always is.
• CFI (Comparative Fit Index)
• Range 0 1 1 is best.
• TLI (Tucker Lewis Index or NNFI)
• Range 0 1 1 is best.
• RMSEA (Root Mean Square Error of Approximation)
• Under .05 is good above .10 is bad.

31
Time-Invariant Covariates

• Age at Baseline
• Education
• Race
• Biological Parents Mental Health Problem
• Lived with both Bio Parents at Age 15
• Number of Previous Relationships
• Baseline SRH

• Considered an Abortion
• Positive Marriage Attitude
• Prenatal Variables (medical care, drug and
  alcohol use, smoking)
• Baseline Marital Status

32
Time-Invariant Covariates
a
ß
33
Figure 5. Mothers Self-Rated Health
Trajectories.
34
Selection Issues

• Intercept
• Third factor is responsible for where people
  start.
• Slope
• Third factor is responsible for where people go.

35
Time-Varying Covariate

• Mental Health Problems
• Range 0-3
• Includes CIDI Major depressive episode, binge
  drinking, and drug use.
• All occurred in the past 12-months.

36
MH 1
MH 3
MH 5
37
Part IV Additional Issues
38
Multi-Models

• Multi-Group
• Growth process may vary for each group.
• Multi-Process
• Models more than one trajectory.

39
Measurement

• Latent Measures (Multiple Indicators)
• Dichotomous/Categorical Variables
• Count Variables
• ZIP Models
• Skewness
• Transform Variable
• Semi-Continuous Growth Model

40
Age-Based Growth Model

• Synthetic cohort
• Sample members may contribute different amounts
  of information at different times.
• Missing Data
• Drop Cases (default)
• Multiple Imputation
• Full Information Maximum Likelihood (FIML)
• Analysis MISSING

41
Mixture Models

• Latent Class Models (LCM/LCA)
• Group membership not known.
• Latent Class Growth Models (LCGM/LCGA)
• Group membership not known and is based on
  trajectory patterns.
• No variation is allowed within latent classes.
• Growth Mixture Models (GMM)
• Group membership is not known and is based on
  trajectory patterns.
• Allows for variation within latent classes.

42
Contact Info

• smeadows_at_princeton.edu